Success is not final, failure is not fatal: it is the courage to continue that counts
March 2023
3.19
Question: $ \lim_{x\to 0} \frac{1}{x} - \frac{1}{e^x-1} $
$$
\begin{equation}
\begin{split}
& \lim_{x\to 0} (\frac{1}{x}- \frac{1}{e^x-1}) \\\\
= & \lim_{x\to 0} \frac{e^x-1-x}{x(e^x-1)} \qquad & (\lim_{x\to 0} e^x-1-x\rightarrow 0, \lim_{x\to 0} x(e^x-1)\rightarrow 0)\\\\
= & \lim_{x\to 0} \frac{[e^x-1-x]'}{[x(e^x-1)]'} \qquad &(\color{red}{L'rule}) \\\\
= & \lim_{x\to 0} \frac{e^x-1}{e^x-1+xe^x} \qquad &(\lim_{x\to 0} e^x-1\rightarrow 0, e^x-1-xe^x\rightarrow 0)\\\\
= & \lim_{x\to 0} \frac{[e^x-1]'}{[e^x-1+xe^x]'} \qquad &(\color{red}{L'rule})\\\\
= & \lim_{x\to 0} \frac{e^x}{e^x+e^x+xe^x} \qquad &(\lim_{x\to 0}xe^x\rightarrow 0) \\\\
= & \lim_{x\to 0} \frac{e^x}{2e^x}\\\\
= & \frac{1}{2}
\end{split}
\end{equation}
$$
3.20
Question:$ \lim_{x\to +\infty} \frac{e^x}{(1+\frac{1}{x})^{x^2}} $
$$
\begin{equation}
\begin{split}
& \lim_{x\to +\infty} \frac{e^x}{(1+\frac{1}{x})^{x^2}} \\\\
= & \lim_{x\to +\infty} e^{x-\ln(1+\frac{1}{x})^{x^2}}\\\\
= & \lim_{x\to +\infty} e^{x-x^2\ln(1+\frac{1}{x})} \qquad &(x\rightarrow \frac{1}{t},t\rightarrow 0 )\\\\
= & \lim_{t\to 0} e^{\frac{1}{t}-\frac{\ln(1+\frac{1}{x})}{t^2}}\\\\
= & \lim_{t\to 0} e^{\frac{t-\ln(1+t)}{t^2}} \qquad &(t-ln(1+t)\rightarrow 0, t^2\rightarrow 0) \\\\
= & \lim_{t\to 0} e^{\frac{[t-\ln(1+t)]'}{[t^2]'}} &(\color{red}{L'Rule})\\\\
= & \lim_{t\to 0} e^{\frac{1-\frac{1}{1+t}}{2t}}\\\\
= & \lim_{t\to 0} e^{\frac{1+t-1}{(1+t)2t}}\\\\
= & \lim_{t\to 0} e^{\frac{t}{2t+2t^2}} \qquad &(t\rightarrow 0,2t+2t^2\rightarrow 0)\\\\
= & \lim_{t\to 0} e^{\frac{[t]'}{[2t+2t^2]'}} \qquad &(\color{red}{L'Rule})\\\\
= & \lim_{t\to 0} e^{\frac{1}{2+2t}}\\\\
= & e^{\frac{1}{2}}
\end{split}
\end{equation}
$$
$$
\begin{equation}
\begin{split}
& \lim_{x\to +\infty} \frac{e^x}{(1+\frac{1}{x})^{x^2}} \\\\
= & \lim_{x\to +\infty} e^{x-x^2\ln(1+\frac{1}{x})}\\\\
= & \lim_{t\to 0} e^{\frac{1}{t}-\frac{1}{t^2}\ln(1+t)} \qquad(\color{red}{Taylor}, \lim_{t\to 0} \ln(1+t)\rightarrow t-\frac{t^2}{2}+o(x^3))\\\\
= & \lim_{t\to 0} e^{\frac{1}{t}-\frac{1}{t^2}(t-\frac{t^2}{2})}\\\\
= & \lim_{t\to 0} e^{\frac{1}{t}-\frac{1}{t}+\frac{1}{2}}\\\\
= & e^{\frac{1}{2}}
\end{split}
\end{equation}
$$
3.21
$$
\begin{equation}
\begin{split}
& \lim_{x\to 0} {(\frac{a^x+b^x+c^x}{3})}^{\frac{1}{x}}\\\\
= &\lim_{x\to 0} e^{\frac{1}{x}\ln(\frac{a^x+b^x+c^x}{3})}\\\\
= &\lim_{x\to 0} e^{\frac{\ln(\frac{a^x+b^x+c^x}{3})}{x}}\qquad (\ln{\frac{a^x+b^x+c^x}{3}}\rightarrow0, x\rightarrow 0,\color{red}{L'Rule})\\\\
= &\lim_{x\to 0} e^{\frac{[\ln(\frac{a^x+b^x+c^x}{3})]'}{[x]'}}\\\\
= &\lim_{x\to 0} e^{\frac{\ln{a}\cdot a^x+\ln{b}\cdot b^x+\ln{c}\cdot c^x}{a^x+b^x+c^x}}\qquad (\lim_{x\to 0} a^x,b^x, c^x\rightarrow 1)\\\\
= & \lim_{x\to 0} e^{\frac{\ln{a}+\ln{b}+\ln{c}}{3}}\\\\
= & (abc)^{\frac{1}{3}}
\end{split}
\end{equation}
$$